Problem # p049

Motion of Rod under Influence of Gravity
The rod of mass m in the figure to the left is in the process of sliding along the wall and floor. Neglect friction. The polar mass moment of inertia of the slender rod of length L and mass m with respect to its center of mass is given by the equation :

= m L² / 12


The purpose of this problem is to ultimately derive the differential equation governing the motion of the rod. You will have to address only the first five questions, see the special comment at the end of question 5.

  1. Carefully sketch the Free Body Diagram of the rod. Label each arrow representing a force and describe in a short sentence what the arrow represents ( who is exerting it onto what ).

  2. Write out the equation for the x-component of the acceleration of the center of mass ( at L/2 from either end ) in terms of mass and x-components of the forces.

  3. Write out the equation for the y-component of the acceleration of the center of mass ( at L/2 from either end ) in terms of mass and y-components of the forces.

  4. Write out the equation for the angular acceleration of the rod in terms of the moment of the forces around the center of mass.

  5. Write out the vector equation for the absolute acceleration of the rod at point B in terms of the absolute acceleration of point A plus the relative acceleration of B as seen by A. Do NOT assume that the angular velocity of the rod is zero !!!

    If you answer the above equations correctly you will get the full amount of credits for this problem. You may keep on going to earn some extra bonus credits.


  6. Write out the equation for the absolute acceleration of the center of mass of the rod in terms of the absolute acceleration of point A and the relative acceleration of the center of mass as seen by A. Do NOT assume that the angular velocity of the rod is zero !!!

  7. By method of substitution ELIMINATE the forces, the 2 components of the acceleration of the center of mass, and finally the acceleration of point A from your collection of equations. You should be left with a single equation relating the angular acceleration d²Θ/dt²=α , the angular velocity dΘ/dt=ω, and the angle Θ to each other. This equation describes ( as a differential equation ) the movement of the rod from some initial condition. Do not attempt to solve this equation; probably, it can be done only on a computer.